Information of high-level, topological contact states is useful and even necessary for a wide range of applications, including many robotic applications. A contact state between two polyhedral objects can be effectively represented as a contact formation in terms of a set of principal contacts between faces, edges, and vertices of the two objects. However, little is done to characterize and represent contact states between curved objects. In order to facilitate the representation of contact states between such objects, we introduce a novel approach to segment the boundary of curved objects based on monotonic changes of curvatures, which we call the curvature monotonic segmentation. We specifically apply this approach to curved 2D and 3D objects with boundary curves or surfaces represented by algebraic polynomials of degrees up to 2. The segmentation yields curvature monotonic faces and edges (or pseudo edges), and vertices (or pseudo vertices). With these faces, (pseudo) edges, and (pseudo) vertices, we effectively extend the concept of contact formation to curved objects to represent high-level, topological contact states between such objects with the same desirable characteristics as the contact formations between polyhedral objects.